Chern-Simons terms in Noncommutative Geometry and its application to Bilayer Quantum Hall Systems

Considering bilayer systems as extensions of the planar ones by an internal space of two discrete points, we use the ideas of Noncommutative Geometry to construct the gauge theories for these systems. After integrating over the discrete space we find an effective $2+1$ action involving an extra complex scalar field, which can be interpreted as arising from the tunneling between the layers. The gauge fields are found in different phases corresponding to the different correlations due to the Coulomb interaction between the layers. In a particular phase, when the radial part of the complex scalar field is a constant, we recover the Wen-Zee model of Bilayer Quantum Hall systems. There are some circumstances, where this radial part may become dynamical and cause dissipation in the oscillating supercurrent between the layers.